To solve the inequality [tex]\(|7x + 3| < 11\)[/tex], we need to consider the definition of absolute value. The absolute value inequality [tex]\(|A| < B\)[/tex] is equivalent to [tex]\(-B < A < B\)[/tex]. Thus, we can rewrite the given inequality without the absolute value as follows:
[tex]\[
-11 < 7x + 3 < 11
\][/tex]
We will solve this compound inequality step by step.
1. Subtract 3 from all parts of the inequality:
[tex]\[
-11 - 3 < 7x + 3 - 3 < 11 - 3
\][/tex]
This simplifies to:
[tex]\[
-14 < 7x < 8
\][/tex]
2. Divide all parts of the inequality by 7 to solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{-14}{7} < \frac{7x}{7} < \frac{8}{7}
\][/tex]
This simplifies to:
[tex]\[
-2 < x < \frac{8}{7}
\][/tex]
Therefore, the solution to the inequality [tex]\(|7x + 3| < 11\)[/tex] is:
[tex]\[
-2 < x < \frac{8}{7}
\][/tex]
